Finding A⁴ + B⁴ Given A² + B² + 1/a² + 1/b² = 4 A Step-by-Step Solution
In this article, we delve into an intriguing mathematical problem involving quadratic expressions and their relationship to higher powers. The problem presents us with the equation a² + b² + 1/a² + 1/b² = 4, where a and b are non-zero real numbers. Our objective is to determine the value of a⁴ + b⁴. This exploration will involve algebraic manipulation, insightful observations, and a step-by-step approach to unravel the solution. We will use fundamental algebraic identities and techniques to transform the given equation into a more manageable form, ultimately leading us to the desired result. This problem not only tests our algebraic skills but also highlights the beauty of mathematical problem-solving, where seemingly complex expressions can be simplified with the right approach. The core idea revolves around recognizing patterns and applying appropriate algebraic identities to reduce the complexity of the equation. By carefully analyzing the given equation and using the properties of squares and reciprocals, we can unlock the hidden relationships between the variables and arrive at the solution. This problem serves as a great example of how mathematical elegance can be found even in seemingly intricate problems.
Problem Statement and Initial Analysis
The given equation a² + b² + 1/a² + 1/b² = 4 is the cornerstone of our problem. We are tasked with finding the value of a⁴ + b⁴, which involves higher powers of a and b. A critical first step is to recognize that the given equation involves both the squares of a and b and the squares of their reciprocals. This suggests that we might be able to manipulate the equation to create perfect squares or other recognizable forms. Our goal is to somehow relate the given equation to an expression involving a⁴ and b⁴. One common strategy in such problems is to try squaring the given equation or rearranging terms to reveal useful identities. We need to explore how the terms a², b², 1/a², and 1/b² interact with each other. A potential pathway is to rearrange the terms and try to form complete squares or use the AM-GM inequality, which often helps in problems involving sums and reciprocals. Another valuable approach is to look for symmetry in the equation. The given equation is symmetric in a and b, meaning that interchanging a and b does not change the equation. This symmetry often hints at possible simplifications or solutions where a and b might have similar properties or values. Understanding the symmetry and the structure of the equation is crucial for selecting the most efficient solution path.
Step-by-Step Solution
Let's embark on the step-by-step solution to find the value of a⁴ + b⁴. Our starting point is the equation:
a² + b² + 1/a² + 1/b² = 4
Step 1: Rearrange and Group Terms
We can rearrange the terms to group a² with 1/a² and b² with 1/b². This grouping is strategic because it allows us to form expressions that resemble the expansion of a square:
(a² + 1/a²) + (b² + 1/b²) = 4
Step 2: Apply Algebraic Manipulation
Now, let's consider the expression (x + 1/x)². Expanding this, we get x² + 2 + 1/x². Notice the similarity to our grouped terms. We can rewrite a² + 1/a² and b² + 1/b² in terms of squares:
a² + 1/a² = (a + 1/a)² - 2 b² + 1/b² = (b + 1/b)² - 2
Substituting these back into our equation, we have:
[(a + 1/a)² - 2] + [(b + 1/b)² - 2] = 4
Step 3: Simplify the Equation
Simplifying the equation, we get:
(a + 1/a)² - 2 + (b + 1/b)² - 2 = 4 (a + 1/a)² + (b + 1/b)² = 8
Step 4: Introduce New Variables
To further simplify the equation, let's introduce new variables:
x = a + 1/a y = b + 1/b
Now our equation becomes:
x² + y² = 8
Step 5: Analyze the Possible Values of x and y
We know that for any real number z, (z - 1/z)² ≥ 0. Expanding this gives z² + 1/z² - 2 ≥ 0, which implies z² + 1/z² ≥ 2. Thus, * (z + 1/z)² = z² + 2 + 1/z² ≥ 4*, so |z + 1/z| ≥ 2. Applying this to our variables a and b, we have:
|a + 1/a| ≥ 2 |b + 1/b| ≥ 2
This means |x| ≥ 2 and |y| ≥ 2. Therefore, x² ≥ 4 and y² ≥ 4.
Step 6: Deduce the Values of x² and y²
Since x² + y² = 8 and both x² and y² are greater than or equal to 4, the only possible solution is x² = 4 and y² = 4. This leads to:
(a + 1/a)² = 4 (b + 1/b)² = 4
Step 7: Find the Values of a and b
Taking the square root of both sides, we get:
a + 1/a = ±2 b + 1/b = ±2
For a + 1/a = 2, we have a² - 2a + 1 = 0, which gives (a - 1)² = 0, so a = 1. For a + 1/a = -2, we have a² + 2a + 1 = 0, which gives (a + 1)² = 0, so a = -1. Similarly, for b + 1/b = 2, we get b = 1, and for b + 1/b = -2, we get b = -1.
Thus, a and b can each be either 1 or -1.
Step 8: Calculate a⁴ + b⁴
Finally, we can calculate a⁴ + b⁴:
If a = 1 and b = 1, then a⁴ + b⁴ = 1⁴ + 1⁴ = 1 + 1 = 2. If a = 1 and b = -1, then a⁴ + b⁴ = 1⁴ + (-1)⁴ = 1 + 1 = 2. If a = -1 and b = 1, then a⁴ + b⁴ = (-1)⁴ + 1⁴ = 1 + 1 = 2. If a = -1 and b = -1, then a⁴ + b⁴ = (-1)⁴ + (-1)⁴ = 1 + 1 = 2.
In all cases, a⁴ + b⁴ = 2.
Conclusion
Through a series of algebraic manipulations and logical deductions, we have successfully found the value of a⁴ + b⁴. The key steps involved rearranging terms, recognizing patterns, introducing new variables, and analyzing the possible values of these variables. The problem showcases the power of algebraic techniques in simplifying complex expressions and arriving at elegant solutions. The final answer is a⁴ + b⁴ = 2. This problem serves as a testament to the beauty and intricacy of mathematics, where careful observation and strategic manipulation can unlock the solutions to seemingly challenging problems. By understanding the underlying principles and applying them methodically, we can unravel the complexities and discover the hidden simplicity within. The journey through this problem has not only provided us with a solution but also enhanced our problem-solving skills and appreciation for mathematical reasoning.
To recap, the main steps we followed to solve the problem are:
- Rearranged and grouped terms to bring similar expressions together.
- Applied algebraic manipulation to rewrite terms as squares and introduce new expressions.
- Simplified the equation to make it more manageable.
- Introduced new variables to further simplify the equation and make it easier to work with.
- Analyzed the possible values of the new variables based on mathematical properties.
- Deducted the specific values of the squared variables.
- Found the values of a and b by solving the simplified equations.
- Calculated a⁴ + b⁴ using the values of a and b.
While the step-by-step solution above is comprehensive, let's briefly discuss potential alternative approaches to this problem. One such approach could involve directly manipulating the given equation to isolate terms involving a⁴ and b⁴. However, this might lead to more complex algebraic expressions. Another approach could involve using inequalities, such as the AM-GM inequality, to establish bounds on the values of a and b. However, the method we used in the step-by-step solution is arguably the most straightforward and efficient way to solve this problem. Exploring different approaches can provide a deeper understanding of the problem and enhance problem-solving skills, even if the primary method is already known.
This problem underscores the importance of having strong problem-solving skills in mathematics. The ability to recognize patterns, manipulate equations, and make logical deductions is crucial for tackling a wide range of mathematical problems. Problem-solving skills are not just about memorizing formulas or procedures; they involve a deeper understanding of mathematical concepts and the ability to apply them creatively. Engaging with problems like this one helps sharpen these skills and build confidence in mathematical abilities. By breaking down complex problems into smaller, manageable steps, we can approach them with clarity and precision. Each step in the solution process is a testament to the power of logical reasoning and the beauty of mathematical problem-solving.
In conclusion, we have successfully navigated through a mathematical challenge, finding the value of a⁴ + b⁴ given the constraint a² + b² + 1/a² + 1/b² = 4. The problem highlighted the significance of algebraic manipulation, logical deduction, and pattern recognition in mathematics. The solution a⁴ + b⁴ = 2 is a testament to the elegance and simplicity that can be found within complex mathematical expressions. By mastering these skills, we can confidently approach a wide range of mathematical problems and appreciate the beauty and power of mathematical reasoning. This journey through the problem has not only provided a solution but has also strengthened our mathematical foundation and problem-solving abilities, preparing us for future challenges in mathematics and beyond.
